Rockfish Historical Harvest Estimates

This project seeks to estimate sport fish harvest and releases of rockfish in Alaska waters by improving on the Howard et al. (2020) methods and expand the time series back to 1977 when the statewide harvest survey (SWHS) was first implemented. This is essentially a Bayesian version of the Howard methods that allows for more appropriate and defensible sharing of information between areas, handles missing data in a more appropriate manor, accurately propagates uncertainty throughout the estimation procedure and replaces the Howard decision tree approach to low sample sizes with a hierarchical model. The methods and results for generating harvest estimates are generally consistent between the Bayesian model and the Howard methods. Harvest estimates are consistent with Howard estimates during contemporary times, but may differ based on more appropriate weighting of SWHS and logbook data, including estimating and correcting bias in the SWHS data.

The Bayesian methods depart from the Howard method in how releases are estimated. The Howard methods assume that the species composition of the harvests are equal to the species composition of released fish, which is clearly contraindicated in the logbook data. For instance, logbook data demonstrates that yelloweye have been retained at high levels up until restrictions were enacted in recent years, whereas pelagic rockfish were released in significant numbers in the past with retention increasing in recent years as they have become more prized by anglers. Recent prohibition on retaining yelloweye in Southeast Alaska highlights the shortcomings of the original Howard assumptions as the species composition of the harvest would indicate that no yelloweye were caught and released during the closure.

The Howard method for estimating releases for private anglers also relied on an expansion of the logbook release estimates based on the ratio of private:guided releases of all rockfish in the SWHS. In addition to the faulty assumptions about species composition, this method ignores potential bias in SWHS estimates of harvests and releases or at least assumes that the bias in release and harvests are the same. As demonstrated in Figure 1, the bias in those two quantities appears to be quite different based on the logbook data. The Bayesian model thus attempts to estimate release probabilities based on the logbook data coupled with bias corrected estimates from the SWHS.

Lastly, the Howard methods were only used on data beginning in 1999 with the advent of the logbook program and estimates of harvests and releases prior to that have been based on linear ramps from 1999 back to the perceived start of the fishery. The Bayesian methods allow us to expand the time series back to 1977 when the SWHS was implemented by leveraging regional data trends in species composition and the proportion of caught rockfish harvested by species and/or species complex. Key advantages of the Bayesian approach are highlighted in table 1.

Table 1. Summary of key improvements in reconstructiing sport fish removals of rockfish using the Bayesian model as compared to the Howard et al. (2020) methods.

Issue	Howard	Bayes
Time series	1999 - present	1977 - present
Bias in SWHS	Not explicitly dealt with. Relies on logbook data and ratios of guided/unguided from SWHS data to estimate unguided releases and harvests.	Explicitly estimates bias in SWHS harvest and release estimates based on logbook data.
Species composition of releases	Assumes that species composition of releases is equal to that of the harvest, which is not evident in the logbook data.	Recognizes different release probabilities by species / species assemblage and estimates it from logbook data and bias corrected SWHS data
Sample size limitations	Uses sample size threshholds such that when areas fall below those threshholds values are borrowed from nearby areas.	Uses a hierarchichacal modelling approach that shares information between areas in the same region. Thus all data is used, even with small sample sizes. This is a more sound method that avoids assumptions and uses all of the data.
Error propogation	Error is propogated when variance estimates are available, but there is uncertainty associated with borrowing values from nearby areas, or the assumption of species compositions being identical in harvest and releases, are not dealt with.	By breaking the assumption that species composition is equal between harvests and releases, uncertainty in the release estimates is more reflective of the fishery. Furthermore, the hyerarchichal approach more accurately captures uncertainy within and between areas within a region.

Data

Harvest data was available for 22 commercial fishing management areas in Southcentral and Southeast Alaska. Areas with negligible rockfish harvest were pooled with adjacent areas for analysis. Specifically the Aleutian and Bering areas were pooled into an area labeled BSAI; the IBS and EKYT were pooled into an area labeled EKYKT; the Southeast, Southwest, SAKPEN and Chignik areas were pooled into an area labeled SOKO2PEN and the Westside and Mainland areas were pooled into an area labeled WKMA.

Stateside Harvest Survey (SWHS)

Statewide harvest survey estimates of rockfish catch and harvest are available for 28 years (1996-2023) for all users and for 13 years (2011-2023) for guided anglers (Figure 0). Additionally, there are overall harvest estimates from 1977- 1995 and release estimates from 1990-1995 that required some partitioning to ascribe to current management units. Harvests in unknown areas were apportioned based on harvest proportions in 1996. Variance estimates are not available for pre-1996 data and as such, the maximum observed coefficient of variation (cv) in each commercial fisheries management unit was applied to the pre-1996 values.

Figure 1.- Data sources for estimating rockfish harvests and releases in ADF&G commercial fisheries management units. Note that initial rockfish harvest estimates were not differentiated into species assemblage or species until 1998 when logbooks began differentiating by pelagic and non-pelagic. Logbooks began to collect data on yelloweye beginning in 2006. Port sampling programs to gather data on species composition of harvests began in 1996 in Southcentral and Kodiak and in 2006 in Southeast.

SWHS estimates are believed to be biased to some degree. These modelling efforts aim to estimate and correct for that bias with the assumption that logbook records are a census of guided harvests and releases.

SWHS Rockfish release estimates are inferred from the difference between catch and harvest estimates.

Adam noted that the first 5 years (23 years counting the historical data) in the SWHS data set for PWSO seem unreasonable (close to zero and not corroborated with logbook estimates). Adam recommended setting these harvests to unknown, but current model development has included the data. Once a satisfactory model has been identified we will exam the effects of censoring the PWSO data.

Creel Surveys

NA

Guide Logbooks

Sport fishing guides have been required to report their harvest of rockfish for 26 years (1998-2023). Reported harvest is also available by assemblage (pelagic vs. non-pelagic). Harvest of yelloweye and “other” (non-pelagic, non-yelloweye) rockfish were reported separately beginning in 2006.

Logbooks also record the number of rockfish released for the same categories. However, the reliability of the release data is somewhat questionable as reported releases are generally far lower than that estimated by the SWHS. As such several treatments of the data are considered.

Logbook versus SWHS estimates

Estimates of guided harvests and releases from the SWHS do not align with the census from charter logbooks. Logbook harvest reports are generally considered reliable and are used to assess the bias in SWHS reports. However, there is even greater disparity between release estimates in the two sources and it is debatable whether logbook releases should be treated as a census. The Howard et al. (2020) methods do treat the logbook release data as “true” and thus are considerably less than would be estimated from the SWHS data.

Figure 2.- SWHS harvest (left) and release (right) estimates from guided trips (x-axis) versus repoted harvests from charter logbooks (y-axis).

A note on model development

To evaluate the discrepancy in apparent bias in harvest and release data, several models were explored to estimate releases during model development. One method (\(LB_{fit}\)) considers the logbook release data to be reliable and a second method (\(LB_{cens}\)) treated the logbook release data as estimates of the minimum released, thus giving more weight to SWHS release estimates. A third method (\(LB_{hyb}\)) is a hybrid approach that treats reported releases of yelloweye as reliable but total rockfish and pelagic rockfish releases as minimums. Model development revealed a tension between the total and pelagic logbook releases and the yelloweye logbook releases. This tensions eventually highlighted the different release/retention probabilities between yelloweye and pelagics in the logbook data and prompted the current approach whereby that probability was calculated for the three main species complexes covered in the data: pelagics, yelloweye, and “other”. The methods described here follow the (\(LB_{fit}\)) formulation. Based on model behavior it is unlikely that the (\(LB_{cens}\)) model would work as there would not be enough data to estimate release probabilities. However, it may be worth running the (\(LB_{hyb}\)) approach as a sensitivity test at the very least.

Composition data

Harvest sampling data exists from Gulf of Alaska areas since 1996 and from Southeast Alaska areas since 2006. Port sampling data is comprised of the number of total rockfish, pelagic and non-pelagic rockfish, black rockfish and yelloweye rockfish. In Southeast Alaska, the number of Demersal Shelf Rockfish (DSR, of which yelloweye are one species) and slope rockfish are also recorded.

Process equations

The true harvest \(H_{ay}\) of rockfish for area \(a\) during year \(y\) is assumed to follow a temporal trend defined by a penalized spline:

\[\begin{equation} \textrm{log}(H_{ay})~\sim~\textrm{Normal}(f(a,y), {\sigma_H}) \end{equation}\]

where \(f(a,y)\) in a p-spline basis with 7 components (knots) and a second degree penalty. The variance, \(\sigma_H\), was given a normal prior with a mean and standard deviation of 0.25 and 1, respectively.

Charter and private harvest \(H_{ayu}\) (where u = 1 for charter anglers and u = 2 for private anglers) is a fraction of total annual harvest in each area:

\[\begin{equation} H_{ay1}~=~H_{ay}P_{(user)ay1}\\H_{ay2}~=~H_{ay}(1-P_{(user)ay1}) \end{equation}\]

where \(P_{(user)ay1}\) is the fraction of the annual harvest in each area taken by charter anglers. \(P_{(user)ay1}\) was modeled hierarchically across years as:

\[\begin{equation} P_{(user)ay1}~\sim~\textrm{beta}(\lambda1_a, \lambda2_a) \end{equation}\]

with non-informative priors on both parameters.

Annual black rockfish harvest \(H_{(black)ayu}\) for each area and user group is:

\[\begin{equation} H_{(black)ayu}~=~H_{ayu}P_{(pelagic)ayu}P_{(black|pelagic)ayu} \end{equation}\]

where \(P_{(pelagic)ayu}\) is the fraction of the annual harvest for each area and user group that was pelagic rockfish and \(P_{(black|pelagic)ayu}\) is the fraction of the annual harvest of pelagic rockfish for each area and user group that was black rockfish.

The southeast region also tracks two other non-pelagic rockfish assemblages, demersal shelf rockfish (DSR, which includes yelloweye) and slope rockfish. For the southeast region the harvest of those two assemblages is thus

\[\begin{equation} H_{(DSR)ayu}~=~H_{ayu}(1-P_{(pelagic)ayu})P_{(DSR|non-pelagic)ayu}\\ H_{(slope)ayu}~=~H_{ayu}(1-P_{(pelagic)ayu})P_{(slope|non-pelagic)ayu}\\ \end{equation}\]

where \(P_{(DSR|non-pelagic)ayu}\) and \(P_{(slope|non-pelagic)ayu}\) are the fractions of the annual harvest of non-pelagic rockfish for each area and user group that were DSR and slope rockfish, respectively.

Annual yelloweye rockfish harvest \(H_{(yelloweye)ayu}\) for each area and user group is calculated differently for central/Kodiak areas and southeast areas. For central and Kodiak areas yelloweye rockfish harvests are calculated as

\[\begin{equation} H_{(yelloweye)ayu}~=~H_{ayu}(1-P_{(pelagic)ayu})P_{(yelloweye|non-pelagic)ayu} \end{equation}\]

where \(P_{(yellow|non-pelagic)ayu}\) is the fraction of the annual harvest of non-pelagic rockfish for each area and user group that was yelloweye rockfish.

For southeast areas yelloweye harvests are a fraction of the DSR harvests such that

\[\begin{equation} H_{(yelloweye)ayu}~=~H_{(DSR)ayu}P_{(yelloweye|DSR)ayu} \end{equation}\]

The composition parameters \(P_{(comp)ayu}\), were modeled using a logistic curve that would allow hindcasting without extrapolating beyond the limit of observed values such that:

\[\begin{equation} \textrm{logit}(P_{(comp)ayu})~=~\beta0_{(comp)ayu} + \frac{\beta1_{(comp)ayu}}{(1 + exp(\beta2_{(comp)ayu}*(y - \beta3_{(comp)ayu})))} + \beta4_{(comp)ayu}*I(u=private)+re_{(comp)ayu} \end{equation}\]

where the \(\beta\) parameters define the intercept, scaling factor, slope, inflection point and private angler effect, respectively, \(y\) is the year index, \(I(u=private)\) is an index variable which is 1 when the user groups is private and 0 otherwise and \(re_{(comp)ayu}\) is a random effect with a non-informative prior. \(\beta\) parameters were modeled hierarchically by region. When \(\beta\) parameters were inestimable as a result of no discernible change in composition over the observed time period. \(\beta1\) (scaling factor) and \(\beta2\) (slope) were fixed to 0 so that the long term mean value was used for hindcasting.

The true number of released rockfish \(R_{ayu}\) were based on the proportion of the total catch harvested, \(pH_{(comp)ayu}\), by area, year, user group and species grouping. Because release data from the SWHS is for all rockfish and the release data from logbooks is only subdivided into pelagics, yelloweye and “other” (non-pelagic, non-yelloweye), we only estimated \(pH_{(comp)ayu}\) for those categories. Thus, converting \(H_{(comp)ayu}\) to total catches by user group, \(C_{(comp)ayu}\), with \(pH_{(comp)ayu}\) results in estimates of total releases such that

\[\begin{equation} R_{(comp)ayu}~=~ C_{(comp)ayu} - H_{(comp)ayu} ~=~ \frac{H_{(comp)ayu}}{pH_{(comp)ayu}} - H_{(comp)ayu} \end{equation}\]

with total releases equal to the sum of the compositional releases. For non-yelloweye DSR and Slope rockfish assemblages in Southeast Alaska \(R_{(DSR)ayu}\) and \(R_{(slope)ayu}\) were estimated from \(R_{(other)ayu}\) using the species composition data from the harvest, thus assuming that slope and DSR assemblages were caught and released at the same rates.

The proportion harvest parameters for \(pH_{(comp)ayu}\) were modeled using a logistic curve that would allow hindcasting based on trends in the data without extrapolating beyond the range of observed values such that

\[\begin{equation} \textrm{logit}(pH_{(pH)ayuc})~=~\beta0_{(pH)ayu} + \frac{\beta1_{(pH)ayuc}}{(1 + exp(\beta2_{(pH)ayuc}*(y - \beta3_{(pH)ayuc})))} + \beta4_{(pH)ayuc}*I(u=private)+re_{(pH)ayuc} \end{equation}\]

A random effect term allowed estimation during the historical period when data is available, but the curve defined by the above equation determined release estimates between 1977 and 1990. As with the compositional trends, \(\beta\) parameters were modeled hierarchically by region. When \(\beta\) parameters were inestimable as a result of no discernable change in harvest probability over the observed time period, \(\beta1\) (scaling factor) and \(\beta2\) (slope) were fixed to 0 so that the long term mean value was applied.

Release mortality (i.e., the number of released rockfish expected to die) was calculated assuming fixed mortality rates developed in each of the regions. Deep water release (DWR) devices were mandated for charter fleets in 2013 and rates were derived from CITATION. Southeast applies basic rates estimated in these studies while Southcentral and Kodiak rates were derived by using historical depth-of-release data to adjust the rates based on area and user group.

The total number of mortalities by year, area, user and species/species assemblage in numbers was calculated by summing harvests and release mortality such that

\[\begin{equation} M_{(comp)ayu}~=~ H_{(comp)ayu} + m_{R-(comp)ayu} * R_{(comp)ayu} \end{equation}\]

where \(m_{R-(comp)ayu}\) is the release mortality rate by year, area, user and species (Figure XX).

Total removals in biomass were converted using the average weight of fish from port sampling?. A minimum sample size per year of X fish was used as the cutoff for including in the data set. Weights were modeled hierarchically to estimate weights in years when data was missing. The total biomass of removals by year, area, user and species was thus

\[\begin{equation} B_{(comp)ayu}~=~ \overline{wt}_{(comp)ayu} * M_{(comp)ayu} \end{equation}\]

where \(\overline{wt}_{(comp)ayu}\) is the mean weight by species, area, user and year.

Observation equations

SWHS estimates of annual rockfish harvest \(\widehat{SWHS}_H{ay}\) were assumed to index true harvest:

\[\begin{equation} \widehat{SWHS}_H{ay}~\sim~\textrm{LogNormal}\left(\textrm{log}(H_{ay}b_{ay}), \sigma_{SWHSHay}^2\right) \end{equation}\]

where bias in the SWHS harvest estimates \(b_H{ay}\) is modeled hierarchically across years as:

\[\begin{equation} b_H{ay}~\sim~\textrm{Normal}(\mu_H{(b)a}, \sigma_H{(b)a}) \end{equation}\]

with non-informative priors on both parameters.

SWHS estimates of guided angler harvest \(\widehat{SWHS}_H{ay1}\) are related to total harvest by:

\[\begin{equation} \widehat{SWHS}_H{ay1}~\sim~\textrm{LogNormal}\left(\textrm{log}(H_{ay1}b_{ay}), \sigma_{SWHS_{ay1}}^2\right) \end{equation}\]

Reported guide logbook harvest \(\widehat{LB}_H{ay}\) is related to true harvest as:

\[\begin{equation} \widehat{LB}_H{ay}~\sim~\textrm{Poisson}(H_{ay1})\\ \widehat{LB}_H{(pelagic)ay}~\sim~\textrm{Poisson}(H_{ay1}P_{(pelagic)ay1})\\ \widehat{LB}_H{(yelloweye)ay}~\sim~\textrm{Poisson}(H_{(yelloweye)ay1})\\ \widehat{LB}_H{(nonpel,nonye)ay}~\sim~\textrm{Poisson}(H_{(nonpel,nonye)ay1})\\ \end{equation}\]

Note that for central and Kodiak areas \(H_{(nonpel,nonye)ay1}\) is equal to the total harvest minus pelagic and yelloweye harvests. For southeast areas \(H_{(nonpel,nonye)ay1}\) is equal to the sum of the DSR and slope harvests minus yelloweye harvests.

SWHS estimates of annual rockfish releases \(\widehat{SWHS}_R{ay}\) were assumed to index true releases in a similar fashion and thus modeled similarly. As such, the release data are related to true releases just as harvests were modeled such that:

\[\begin{equation} \widehat{LB}_R{ay}~\sim~\textrm{Poisson}(R_{ay1})\\ \widehat{LB}_R{(pelagic)ay}~\sim~\textrm{Poisson}(R_{ay1}P_{(pelagic)ay1})\\ \widehat{LB}_R{(yelloweye)ay}~\sim~\textrm{Poisson}(R_{(yelloweye)ay1})\\ \widehat{LB}_R{(nonpel,nonye)ay}~\sim~\textrm{Poisson}(R_{(nonpel,nonye)ay1})\\ \end{equation}\]

Because logbook release data is more questionable and demonstrates greater disagreement with SWHS estimates (Figure 1), a second approaches was considered that loosened the assumption that logbook releases were a census. Methods explored to develope \(LB_{hyb}\) and \(LB_{cens}\) models are detailed at the end of this section.

SWHS estimates of guided angler release \(\widehat{SWHS}_R{ay1}\) is modeled the same as harvests.

SWHS release bias was modeled independently of the harvest bias \(b_H{ay}\) such that

\[\begin{equation} b_R{ay}~\sim~\textrm{Normal}(\mu_R{(b)a}, \sigma_R{(b)a}) \end{equation}\]

where bias in the SWHS release estimates \(b_R{ay}\) is modeled hierarchically across years as:

\[\begin{equation} b_R{ay}~\sim~\textrm{Normal}(\mu_R{(b)a}, \sigma_R{(b)a}) \end{equation}\]

with non-informative priors on both parameters.

The number of pelagic rockfish sampled in harvest sampling programs \(x_{(pelagic)ayu}\) follow a binomial distribution:

\[\begin{equation} x_{(pelagic)ayu}~\sim~\textrm{Binomial}(P_{(pelagic)ayu}, N_{ayu}) \end{equation}\]

where \(N_{ayu}\) is the total number of rockfish sampled in area \(a\) during year \(y\) form user group \(u\). The number of black rockfish sampled in harvest sampling programs was thus a proportion of the pelagic harvests

\[\begin{equation} x_{(black)ayu}~\sim~\textrm{Binomial}(P_{(black)ayu}, N_{ayu}^{pel}) \end{equation}\]

Yelloweye rockfish in Southcentral and Kodiak were modeled similarly as a proportion of the total number of non-pelagics such that

\[\begin{equation} x_{(yellow_{R2})ayu}~\sim~\textrm{Binomial}(P_{(yellow_{R2})ayu}, N_{ayu}^{nonpel}) \end{equation}\]

Southeast areas have several other non-pelagic groupings such that DSR and slope rockfish are a proportion of non-pelagics

\[\begin{equation} x_{(DSR)ayu}~\sim~\textrm{Binomial}(P_{(DSR)ayu}, N_{ayu}^{nonpel}) \end{equation}\]

and

\[\begin{equation} x_{(slope)ayu}~\sim~\textrm{Binomial}(P_{(slope)ayu}, N_{ayu}^{nonpel}) \end{equation}\]

with yelloweye in southeast a proportion of the DSR harvest

\[\begin{equation} x_{(yellow_{R1})ayu}~\sim~\textrm{Binomial}(P_{(yellow_{R1})ayu}, N_{ayu}^{DSR}). \end{equation}\].

Kodiak has limited port sampling beyond the main harbors but has a robust hydroacoustic survey that is used to quantify black rockfish abundance across the management area and uses stereocameras to derive species compositions of the hydroacoustic data. This data was used as supplementary data to further inform the model to the proportion of pelagic rockfish that are black in Kodiak areas. Angler landings in Kodiak show a higher proportion of black rockfish relative to the hydroacoustic survey and thus the proportion of black rockfish in the hydroacoustic sample related to the true proportion such that

\[\begin{equation} P_{(black|pelagic)ayu}^{HA} ~\sim~ P_{(black|pelagic)ayu} + ae_{au} \end{equation}\].

where \(ae_{au}\) is the angler effect for each area and user group modeled hierarchically around a mean of 0. Predicted \(P_{(black|pelagic)ayu}^{HA}\) assumed a beta distribution such that

\[\begin{equation} P_{(black|pelagic)ayu}^{HA} ~\sim~ beta(\alpha_{HA},\beta_{HA}) \end{equation}\]

where

\[\begin{equation} \alpha_{HA} ~=~ (P_{(black|pelagic)ayu}^{HA})^2 * \frac{1 - P_{(black|pelagic)ayu}^{HA}}{\frac{var_{P_{HA}}-1}{P_{(black|pelagic)ayu}^{HA}}}, \end{equation}\]

\[\begin{equation} \beta_{HA} ~=~ (\alpha_{HA}) * \frac{1}{P_{(black|pelagic)ayu}^{HA} - 1}, \end{equation}\]

\[\begin{equation} var_{P_{HA}} ~=~ (P_{(black|pelagic)ayu}^{HA} * cvP_{(black|pelagic)ayu}^{HA})^2 \end{equation}\]

where \(cvP_{(black|pelagic)ayu}^{HA}\) is the coefficient of variation for the hydroacoustic proportions

\[\begin{equation} cvP_{(black|pelagic)ayu}^{HA} ~=~ \frac{\sqrt{varP_{(black|pelagic)ayu}^{HA}}}{P_{(black|pelagic)ayu}^{HA}} \end{equation}\]

and the variance is approximated using the XXXX method as

\[\begin{equation} varP_{(black|pelagic)ayu}^{HA} ~=~ (\frac{1}{n_{pel}})^2 * varN_{black} + (\frac{n_{black}}{n_{pel}^2}) * varN_{pel} \end{equation}\]

where \(varN_{black}\) and \(varN_{black}\) are the variance of the estimated number of black and pelagic rockfish in the hydroacoustic survey, respectively (CITATION).

The average weight of rockfish by species, user, area and year was modeled hierarchically at several levels within regions such that

\[\begin{equation} wt_{(comp)ayu} ~\sim~ Normal(wt_{(comp)au},\sigma_{wt_{(comp)au}}) ~\sim~ Normal(wt_{(comp)a},\sigma_{wt_{(comp)a}}) ~\sim~ Normal(wt_{(comp)region},\sigma_{wt_{(comp)region}}) \end{equation}\]

where region refers to Kodiak, Southcentral and Southeast. Mean weights and variance were calculated as XXX.

Alternative likelihoods for release estimates

To loosen the assumption that logbook release data are an effective census of true releases I explored models that treated logbook release estimates as a lower bound on the estimate of true releases. In a hybrid approach yelloweye and non-pelagic releases are regarded as a reliable census (given the emphasis and ease of recording these fish) but censors the pelagic and total rockfish release estimates (where censoring implies NA values) such that

\[\begin{equation} \text{censored} \widehat{LB}_R{ay}~\sim~\textrm{LogNormal}\left(\log(R_{ay}), 1\right)\text{T}\left(\widehat{LB}_R{ay}, \infty\right) \\ \text{censored} \widehat{LB}_R{(pelagic)ay}~\sim~\textrm{LogNormal}\left(\log(R_{(pelagic)ay}), 1\right)\text{T}\left(\widehat{LB}_R{(pelagic)ay}, \infty\right) \\ \widehat{LB}_R{(yelloweye)ay}~\sim~\textrm{Poisson}(R_{(yelloweye)ay1})\\ \widehat{LB}_R{(nonpel,nonye)ay}~\sim~\textrm{Poisson}(R_{(nonpel,nonye)ay1})\\ \end{equation}\]

This model formulation failed such that there was not enough data to inform pelagic releases and the values did not seem valid. A second approach is being explored that fits the censored data using a lognormal distribution centered around the logbook release value, but also with a lower bound equal to the number of recorded releases such that

\[\begin{equation} \widehat{LB}_R{ay}~\sim~\textrm{LogNormal}\left(\log(R_{ay}), \sigma_{Ray1}^2\right)\text{T}\left(\widehat{LB}_R{ay}, \infty\right) \\ \widehat{LB}_R{(pelagic)ay}~\sim~\textrm{LogNormal}\left(\log(R_{(pelagic)ay}), \sigma_{Ray1}^2\right)\text{T}\left(\widehat{LB}_R{(pelagic)ay}, \infty\right) \\ \widehat{LB}_R{(yelloweye)ay}~\sim~\textrm{Poisson}(R_{(yelloweye)ay1})\\ \widehat{LB}_R{(nonpel,nonye)ay}~\sim~\textrm{Poisson}(R_{(nonpel,nonye)ay1})\\ \end{equation}\]

Logbook data is assumed to be a census and as such there is no estimate of uncertainty. As of this writing, several methods are being examined for how to treat \(\sigma_{Ray1}^2\). Models are being run that attempt to allow the model to estimate \(\sigma_{Ray1}^2\) with priors. A simple model applies a uniform prior (0.1,50) to \(\sigma_{Ray1}^2\). A hierarchichal approach based on regions is also being examined whereby \(\sigma_{Ray1}^2\) is lognormally distributed around hyper priors \(\mu_{\sigma_R}\) and \(\sigma_{\sigma_R}\). Initial efforts have applied a uniform prior on \(\mu_{\sigma_R}\) between 1 and 50 and on \(\sigma_{\sigma_R}\) between 0 and 10.

Priors.

Priors range from uninformative to very informative or fixed. Priors for compositional logistic parameters are in Table 2 and proportion harvest logistic parameters are in Table 3. Until I figure out how to make a nice table in Rmarkdown, please refer to the attached spreadsheet and comp and harvp tabs.

Unresolved issues and outstanding questions:

1. Reliability of unguided release estimates: These estimates have the least information feeding them and rely on the bias-corrected SWHS release estimates of all rockfish and the trends in release probability evident in the logbook data. The \(\beta4\) term that estimates the guided/unguided effect was given a very informative prior that tied the release probability of private anglers tightly to that of the charter fleet. The model is then trying to balance the three species complex estimates (pelagic, yelloweye and other) so that they sum to the total unguided releases estimated from the bias corrected SWHS data. For the most part this seems reasonable and appears to work, but there are certain areas where the estimates are “wonky”:

   1. Total rockfish releases more or less align with the total releases estimated with the Howard methods. Presumably, much of the discrepancy results from the substantial bias in release estimates from the SWHS. Interestingly, the logbook data indicates that the SWHS underestimates harvests but overestimates releases by a significant factor (Figure 23 and 24 below).
   2. In general, release estimates of black rockfish are substantially lower than those calculated using the Howard methods. Presumably, much of this derives from the bias correction of the SWHS release estimates.
   3. Yelloweye release estimates also differ considerably from the Howard estimates, but unlike black rockfish are sometimes lower and sometimes higher. Two areas in particular are a little head scratching. Yelloweye releases in the Kodiak Northeast area in particular are significantly lower than for guided anglers with the same pattern evident in Cook Inlet to a lesser extent. Cook Inlet yelloweye numbers are very small, so this is a sample size issue with little consequence. The cause of the Kodiak northeast estimates is not clear to me at this point, but the model estimates the proportion harvested by unguided anglers to be much lower than that of guided anglers, even with the informative prior on \(\beta4\). This must be a product of the bias corrected SWHS release estimates and how the model is partitioning that estimate into the 3 species complexes, but itis a bit a of head scratcher.

2. Proportion guided estimates: There is not much data on this proportion prior to 2011 and it is not modeled with any sort of trend as was done for species composition and harvest proportions. With the exception of Cook Inlet and North Gulf Coast areas, there is little, if any, trend apparent in the data and perhaps this approach is the best available given the data available. However, if there are data sources somewhere that could inform this part of the model they could be incorporated.

3. Prior choices in general need to be vetted. The priors on the logistic curves are fairly informed in an effort to achieve the desired shapes for hindcasting. Ideally, sensitivity testing would occur but the model is very slow to converge. The beta parameters on the logistic curves have required a lot of work on the priors to reach convergence.

4. Proportion harvest estimates for non-pelagic, non-yelloweye in Kodiak WKMA: I need to adjust the prior on the inflection point, \(\beta3\), so that it is forced to occur after 2006. Right now the model is estimating inflection in two Kodiak areas before that point where there is no data to justify a shift. The current inflection is a result of the hierachichal model.

5. Proportion pelagic in PWS and CSEO: The parameters for these particular proportions are very slow to converge. For the CSEO, the estimates of the \(\beta\) parameters are similar to the other Southeast areas, but the mixing is poor over the length of the chains. In this case I think they will ultimately converge with a very long model run and the shape of the curve in the model output looks acceptable. For the two PWS areas the model seems to struggle with the disparate proportional data from the logbook and the port sampling. There is some wandering in the chains of the \(\beta0\) and \(\beta1\) terms and spikiness in the \(\beta2\) terms. I’ve been working on constraining the hyperpriors for PWS \(beta2\). Similar to CSEO, it may just entail a very long model run to reach convergence, but the shape of the curves looks reasonable.

Next steps:

Once the model is finalized, harvest and release numbers need to be converted into biomass removals. This is a two step process where release mortality estimates are applied to the release estimates to estimate the number of released rockfish that do not survive. This is based on studies and will reflect the values that the department has been using with the Howard methods. Region 2 (both Southcentral and Kodiak) have release-at-depth estimates from a number of years that they apply across all years and then calculate mortality rates based on those estiates. Southease does not have release-at-depth data and simply applies an assumed rate based on research.

Once release mortality is calculated average weight data is applied to convert numbers to biomass. The plan is to incorporate all of this into the model to propogate uncertainty into the posteriors. However, the model already takes a long time to run and I may explore a simpler approach using the posteriors from the numbers model to speed up processing.

Results

• Model: LB_fit_3pH

• Model Run: rf_harvest_est_kha_rm_wt_thru2023_1500000_7kn_2025-05-25 (Naming convention: modelname_thruYEAR_iters_rundate)

• Proportion of parmaters converged (Rhat <= 1.1): 0.9947

• See end of document for summary of unconverged parameters.

Figure X.- Rhat values and proportion of parameters that converged (Rhat < 1.1.)

Estimate comparison

Since previous estimates of rockfish harvest have been produced these first 3 graphs will be used to show how the modeled estimates compare to the estimates produced earlier. For total rockfish the estimates are in general agreement although differences are noted. These estimates should be more reliable because they include both SWHS and guide logbook data, handle variance more appropriately, use hierarchical distributions when data is missing, directly consider observation error and are produced using reproducible research.

Figure 3.- Total rockfish harvests 1996-2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 3.- Total rockfish releases 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Notes from Adam: When looking at only black rockfish the most significant differences are for the Prince William Sound Inside area. I did not spend a great deal of time tracking this down although it looks like the previous version used bad values for \(P_{(black)ayu}\) for at least unguided anglers. For the moment I would ignore the results for BSIA and SOKO2SAP. I think it is possible to give approximate values for these areas but it will require a little more coding which I have yet to do.

Figure 4.- Black rockfish harvests 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

And black rockfish releases…

Figure 5.- Black rockfish releases 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 6.- Yellow rockfish harvests 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 7.- Yellow rockfish releases 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 8.- DSR rockfish (including yelloweye) harvests 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 9.- DSR rockfish releases (including yelloweye) 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 11.- Slope rockfish harvests 1996-2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 12.- Slope rockfish releases 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Total Biomass Removal Estimates

Figure 13.- Black rockfish estimated total removals in lbs in 1996–2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

Figure 14.- Yellow rockfish estimated total removals in lbs in 1996–2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

Figure 15.- Pelagic rockfish estimated total removals in lbs in 1996–2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

Figure 16.- Non-yelloweye, demersal shelf rockfish estimated total removals in lbs in 1996-2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

Figure 17.- Slope rockfish estimated total removals in lbs in 1996-2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

Model fit

Logbook residuals

Figure 18.- Residuals from logbook harvests.

SWHS residuals

Figure 19.- Residuals from SWHS harvests.

Figure 20.- Residual of SWHS releases.

Parameter estimates

P(Charter)

These histograms show the posterior distribution of the mean percent of rockfish harvested by the charter fleet.

Figure 21.- Mean percent of harvest by charter anglers.

When considered annually we see the percent of rockfish harvested by the charter fleet follows our data fairly well although the model smooths out the changes and we just do not have much information about this ratio. Prior to 2011 the percent charter is confounded with SWHS bias and should be mostly discounted.

Figure 22.- Annual estimates of the percent of harvest by charter anglers for 16 commerical fishing manamgent areas, 1996-2023.

P(Harvest)

These plots show the fitted logistic line to the proportion of caught rockfish that are harvested. These estimates are used for hindcasting catch estimates based on the harvest data in early years when catch estimates are unavailable.

Figure 23.- Annual proportion of pelagic rockfish catch that was harvested. Note that pre-1990 estimates are used to estimate catch in these years when catch estimates are not available.

Figure 24.- Annual proportion of yelloweye rockfish catch that was harvested. Note that pre-1990 estimates are used to estimate catch in these years when catch estimates are not available.

Figure 25.- Annual proportion of non-pelagic, non-yelloweye rockfish catch that was harvested. Note that pre-1990 estimates are used to estimate catch in these years when catch estimates are not available. Note, that this is not estimated for Southeast areas because non=pelagics are divided between DSR (including yelloweye) and Slope species.

## NULL

## NULL

SWHS bias

Figure 23 shows the mean estimate for SWHS bias in harvests and releases. Cook Inlet, North Gulf Coast and North Southeast Inside all look pretty good while most other areas have substantial bias. Prince William Sound Inside has the largest bias. Bias in release estimates is substantial and whereas the SWHS appears to underestimate harvests, it appears to greatly overestimates releases by a factor of 2 or more in most areas as derived from logbook reported releases.

Figure 28.- Mean SWHS bias for harvests and catches. Note that a bias < 1 indicates that the SWHS underestimates the true value and bias > 1 indicates the survey overestimates the true value.

Our estimates of SWHS harvest bias track observations fairly well when he have guided harvest estimates. The estimates of release bias in the SWHS data track observed patterns to an extent, but appear to smooth these more volatile disagreements with the logbook data. Adam postulated in his initial start on this that some of this could be the result of the estimates of the proportion guided. This value was not modelled with a trend and thus applies a constant estimate when hindcasting. Data on these relationships could greatly improve this model.

Figure 29.- Annual estimates of SWHS bias in harvests and releases for 16 commerical fishing manamgent areas, 1996-2023. Note that a bias < 1 indicates that the SWHS underestimates the true value and bias > 1 indicates the survey overestimates the true value.

P(pelagic)

We model the percentage of pelagic rockfish in the harvest because we have the information for charter anglers (via logbooks) starting in 1998. Other than looking at the model estimates you can use Figure 25 to compare the two data streams for pelagic rockfish harvest. In general they are in agreement with major exceptions in Price William Sound inside, Prince William Sound outside (early in the time series) and South Southeast inside.

Figure 30.- Annual estimates of the percent of the sport harvest that was pelagic rockfish for 16 commerical fishing manamgent areas, 1996-2023.

P(black|pelagic)

Note that in Southeast Alaska we only have composition data starting in 2006. Tania dug up old SE data, but it did not provide any useful data for species apportionment. For the most part, P(black|pelagic) is relatively constant across areas, with the exception of Cook Inlet and NSEI in Southeast AK. It may be worth discussing whether the shifts in those areas is a result of improved or changing species identification rather than actual shift in the species composition of the catch.

Figure 31.- Annual estimates of the percent of the sport harvest of pelagic rockfish that were black rockfish for 16 commerical fishing manamgent areas, 1996-2023. Kodiak panels include data from a hydroacoustic survey and the proportion of pelagic rockfish that are black in those areas (red) and the adjusted proportions based on obseved harvests for charter (blue) and private (cyan) users.

P(yelloweye|non-pelagic / yelloweye|DSR)

Figure 32.- Annual estimates of the percent of the sport harvest of non-pelagic rockfish that were yelloweye rockfish for 16 commerical fishing manamgent areas, 1996-2023. Note that P(yelloweye) is the the proportion relative to non-pelagics for Central and Kodiak areas but is relative to DSR for Southeast areas.

P(DSR|non-pelagic)

Figure 33.- Annual estimates of the percent of the sport harvest of non-pelagic rockfish that were DSR rockfish for 6 Southeast commerical fishing manamgent areas, 1996-2023.

P(slope|non-pelagic)

Figure 34.- Annual estimates of the percent of the sport harvest of non-pelagic rockfish that were slope rockfish for 6 southeast commerical fishing manamgent areas, 1996-2023. Note that P(yelloweye) is the the proportion relative to non-pelagics for Central and Kodiak areas but is relative to DSR for Southeast areas.

P(slope|non-pelagic & non-yellowye) For release estimates

Figure 35.- Annual estimates of the percent of the sport non-pelagic, non-yelloweye releases that were slope rockfish for 6 southeast commerical fishing manamgent areas, 1996-2023.

Weight Fits

Figure 36.- Mean weights of black rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.

Figure 37.- Mean weights of yelloweye rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.

Figure 38.- Mean weights of non-black, pelagic rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.

Figure 39.- Mean weights of non-yelloweye, demersal shelf rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.

Figure 40.- Mean weights of slope rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.

### Summary of unconverged parameters:

Table 1. Summary of unconverged parameters including the number (n) and the average Rhat from the unconverged parameters.

parameter n badRhat_avg beta1_pelagic 3 1.399997 beta0_pelagic 3 1.377373 beta2_pelagic 2 1.353275 beta4_yellow 4 1.204574 beta4_pelagic 4 1.191563

parameter n badRhat_avg beta1_pH 4 1.190083 beta2_pH 1 1.183916 beta3_pelagic 1 1.169991 beta3_yellow 1 1.145337 beta2_yellow 1 1.131006

Table 2. Summary of unconverged parameters by area

	BSAI	CSEO	eastside	NSEI	NSEO	PWSI	PWSO	SOKO2SAP	WKMA
beta0_pelagic	0	1	0	0	0	1	1	0	0
beta1_pelagic	0	1	0	0	0	1	1	0	0
beta1_pH	1	0	0	0	0	1	1	1	0
beta2_pelagic	0	1	0	0	1	0	0	0	0
beta2_pH	0	0	0	0	0	0	0	0	1
beta2_yellow	0	0	0	1	0	0	0	0	0
beta3_pelagic	0	1	0	0	0	0	0	0	0
beta3_yellow	0	0	0	0	1	0	0	0	0
beta4_pelagic	0	1	1	0	0	0	0	1	1
beta4_yellow	1	0	1	0	0	0	0	1	1

Parameter estimates:

Summary Table of Parameter Estimates

Parameter	mean	sd	Lower_CI	Median	Upper_CI
mu_bc_H[1]	-0.128	0.070	-0.256	-0.131	0.017
mu_bc_H[2]	-0.096	0.045	-0.173	-0.100	0.003
mu_bc_H[3]	-0.433	0.072	-0.570	-0.435	-0.285
mu_bc_H[4]	-0.984	0.194	-1.370	-0.981	-0.609
mu_bc_H[5]	0.921	0.901	-0.152	0.743	3.128
mu_bc_H[6]	-2.163	0.324	-2.788	-2.173	-1.524
mu_bc_H[7]	-0.457	0.108	-0.678	-0.455	-0.253
mu_bc_H[8]	0.241	0.364	-0.357	0.206	1.049
mu_bc_H[9]	-0.294	0.132	-0.553	-0.292	-0.031
mu_bc_H[10]	-0.107	0.070	-0.237	-0.109	0.038
mu_bc_H[11]	-0.123	0.037	-0.194	-0.123	-0.050
mu_bc_H[12]	-0.254	0.107	-0.477	-0.249	-0.059
mu_bc_H[13]	-0.135	0.079	-0.288	-0.136	0.025
mu_bc_H[14]	-0.299	0.098	-0.496	-0.300	-0.122
mu_bc_H[15]	-0.345	0.049	-0.441	-0.346	-0.249
mu_bc_H[16]	-0.260	0.386	-0.911	-0.292	0.540
mu_bc_R[1]	1.296	0.143	1.031	1.291	1.593
mu_bc_R[2]	1.453	0.093	1.272	1.454	1.638
mu_bc_R[3]	1.391	0.144	1.102	1.393	1.674
mu_bc_R[4]	0.917	0.207	0.482	0.929	1.303
mu_bc_R[5]	1.175	0.471	0.244	1.186	2.056
mu_bc_R[6]	-1.596	0.422	-2.427	-1.597	-0.754
mu_bc_R[7]	0.461	0.212	0.020	0.474	0.860
mu_bc_R[8]	0.558	0.200	0.154	0.563	0.936
mu_bc_R[9]	0.347	0.203	-0.072	0.358	0.700
mu_bc_R[10]	1.295	0.135	1.011	1.299	1.558
mu_bc_R[11]	1.036	0.098	0.844	1.034	1.234
mu_bc_R[12]	0.822	0.202	0.431	0.819	1.220
mu_bc_R[13]	1.028	0.104	0.820	1.026	1.230
mu_bc_R[14]	0.894	0.143	0.619	0.895	1.171
mu_bc_R[15]	0.783	0.109	0.566	0.783	1.000
mu_bc_R[16]	1.090	0.127	0.840	1.093	1.343
tau_pH[1]	5.165	0.445	4.340	5.140	6.103
tau_pH[2]	2.031	0.226	1.609	2.024	2.502
tau_pH[3]	2.126	0.217	1.729	2.115	2.572
beta0_pH[1,1]	0.547	0.177	0.189	0.554	0.886
beta0_pH[2,1]	1.377	0.172	1.030	1.380	1.711
beta0_pH[3,1]	1.429	0.192	1.002	1.444	1.762
beta0_pH[4,1]	1.564	0.224	1.066	1.583	1.954
beta0_pH[5,1]	-0.865	0.293	-1.472	-0.846	-0.350
beta0_pH[6,1]	-0.717	0.388	-1.619	-0.660	-0.088
beta0_pH[7,1]	-0.523	0.450	-1.475	-0.493	0.351
beta0_pH[8,1]	-0.650	0.275	-1.251	-0.628	-0.181
beta0_pH[9,1]	-0.666	0.288	-1.279	-0.633	-0.177
beta0_pH[10,1]	0.229	0.193	-0.150	0.232	0.594
beta0_pH[11,1]	-0.078	0.162	-0.406	-0.072	0.228
beta0_pH[12,1]	0.496	0.187	0.128	0.495	0.843
beta0_pH[13,1]	0.015	0.155	-0.301	0.018	0.312
beta0_pH[14,1]	-0.318	0.169	-0.672	-0.311	-0.006
beta0_pH[15,1]	-0.028	0.182	-0.394	-0.022	0.308
beta0_pH[16,1]	-0.484	0.361	-1.385	-0.423	0.052
beta0_pH[1,2]	2.847	0.165	2.508	2.855	3.157
beta0_pH[2,2]	2.890	0.134	2.624	2.889	3.146
beta0_pH[3,2]	3.136	0.149	2.860	3.132	3.448
beta0_pH[4,2]	2.952	0.133	2.683	2.949	3.218
beta0_pH[5,2]	4.878	1.498	2.971	4.550	8.654
beta0_pH[6,2]	3.112	0.203	2.715	3.110	3.530
beta0_pH[7,2]	1.837	0.195	1.453	1.836	2.228
beta0_pH[8,2]	2.866	0.172	2.537	2.871	3.196
beta0_pH[9,2]	3.443	0.223	3.011	3.441	3.883
beta0_pH[10,2]	3.753	0.201	3.364	3.755	4.148
beta0_pH[11,2]	-4.827	0.294	-5.418	-4.823	-4.255
beta0_pH[12,2]	-4.779	0.400	-5.607	-4.764	-4.009
beta0_pH[13,2]	-4.566	0.390	-5.320	-4.581	-3.759
beta0_pH[14,2]	-5.595	0.477	-6.554	-5.582	-4.727
beta0_pH[15,2]	-4.284	0.329	-4.918	-4.276	-3.630
beta0_pH[16,2]	-4.839	0.374	-5.634	-4.833	-4.120
beta0_pH[1,3]	-0.142	0.716	-1.722	-0.057	1.016
beta0_pH[2,3]	2.185	0.162	1.867	2.183	2.513
beta0_pH[3,3]	2.529	0.151	2.244	2.530	2.832
beta0_pH[4,3]	2.960	0.163	2.630	2.963	3.276
beta0_pH[5,3]	2.166	1.436	0.399	1.853	5.755
beta0_pH[6,3]	1.001	0.489	-0.151	1.034	1.877
beta0_pH[7,3]	0.625	0.173	0.302	0.620	0.967
beta0_pH[8,3]	0.307	0.187	-0.059	0.311	0.659
beta0_pH[9,3]	-0.624	0.372	-1.512	-0.593	0.012
beta0_pH[10,3]	0.474	0.375	-0.409	0.507	1.095
beta0_pH[11,3]	-0.165	0.323	-0.760	-0.180	0.511
beta0_pH[12,3]	-0.852	0.348	-1.584	-0.827	-0.248
beta0_pH[13,3]	-0.125	0.311	-0.716	-0.131	0.491
beta0_pH[14,3]	-0.272	0.262	-0.779	-0.276	0.255
beta0_pH[15,3]	-0.688	0.302	-1.285	-0.683	-0.110
beta0_pH[16,3]	-0.391	0.294	-0.956	-0.390	0.216
beta1_pH[1,1]	3.080	0.313	2.530	3.049	3.743
beta1_pH[2,1]	2.140	0.263	1.671	2.126	2.688
beta1_pH[3,1]	1.973	0.323	1.453	1.940	2.672
beta1_pH[4,1]	2.381	0.346	1.830	2.330	3.195
beta1_pH[5,1]	2.303	0.364	1.709	2.281	3.034
beta1_pH[6,1]	3.747	0.970	2.361	3.563	6.031
beta1_pH[7,1]	2.633	0.874	0.981	2.567	4.514
beta1_pH[8,1]	4.004	0.999	2.644	3.768	6.537
beta1_pH[9,1]	2.375	0.422	1.724	2.321	3.331
beta1_pH[10,1]	2.397	0.270	1.905	2.390	2.962
beta1_pH[11,1]	3.258	0.208	2.862	3.256	3.667
beta1_pH[12,1]	2.536	0.218	2.130	2.532	2.967
beta1_pH[13,1]	2.961	0.223	2.542	2.946	3.428
beta1_pH[14,1]	3.421	0.220	3.017	3.414	3.886
beta1_pH[15,1]	2.528	0.230	2.099	2.523	2.999
beta1_pH[16,1]	4.124	0.643	3.210	4.018	5.738
beta1_pH[1,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[2,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[3,2]	0.000	0.004	0.000	0.000	0.001
beta1_pH[4,2]	0.000	0.011	0.000	0.000	0.001
beta1_pH[5,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[6,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[7,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[8,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[9,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[10,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[11,2]	6.665	0.325	6.024	6.666	7.304
beta1_pH[12,2]	6.449	0.466	5.574	6.437	7.448
beta1_pH[13,2]	6.948	0.433	6.074	6.952	7.788
beta1_pH[14,2]	7.237	0.497	6.332	7.232	8.242
beta1_pH[15,2]	6.764	0.365	6.046	6.765	7.492
beta1_pH[16,2]	7.432	0.417	6.655	7.415	8.296
beta1_pH[1,3]	4.689	1.651	2.140	4.444	8.075
beta1_pH[2,3]	0.000	0.000	0.000	0.000	0.000
beta1_pH[3,3]	0.000	0.000	0.000	0.000	0.000
beta1_pH[4,3]	0.000	0.000	0.000	0.000	0.000
beta1_pH[5,3]	3.695	5.628	0.852	2.774	12.240
beta1_pH[6,3]	3.204	5.772	0.402	2.613	8.008
beta1_pH[7,3]	0.000	0.000	0.000	0.000	0.000
beta1_pH[8,3]	2.746	0.343	2.067	2.740	3.426
beta1_pH[9,3]	2.753	0.447	2.016	2.724	3.768
beta1_pH[10,3]	2.904	0.441	2.148	2.865	3.914
beta1_pH[11,3]	2.745	0.383	1.980	2.751	3.505
beta1_pH[12,3]	4.097	0.435	3.286	4.074	4.971
beta1_pH[13,3]	1.705	0.339	1.032	1.713	2.342
beta1_pH[14,3]	2.517	0.333	1.853	2.519	3.165
beta1_pH[15,3]	1.976	0.331	1.296	1.977	2.615
beta1_pH[16,3]	1.800	0.323	1.143	1.796	2.433
beta2_pH[1,1]	0.482	0.123	0.289	0.467	0.762
beta2_pH[2,1]	0.592	0.307	0.262	0.522	1.364
beta2_pH[3,1]	0.671	0.493	0.212	0.562	1.886
beta2_pH[4,1]	0.484	0.205	0.209	0.448	0.951
beta2_pH[5,1]	1.441	0.956	0.244	1.317	3.663
beta2_pH[6,1]	0.187	0.062	0.099	0.178	0.333
beta2_pH[7,1]	0.027	0.292	0.000	0.000	0.143
beta2_pH[8,1]	0.247	0.108	0.129	0.229	0.464
beta2_pH[9,1]	0.418	0.205	0.164	0.383	0.876
beta2_pH[10,1]	0.614	0.283	0.294	0.558	1.290
beta2_pH[11,1]	0.785	0.200	0.482	0.752	1.241
beta2_pH[12,1]	1.381	0.475	0.750	1.285	2.629
beta2_pH[13,1]	0.746	0.235	0.408	0.708	1.292
beta2_pH[14,1]	0.848	0.223	0.529	0.812	1.405
beta2_pH[15,1]	0.819	0.303	0.415	0.757	1.552
beta2_pH[16,1]	0.373	0.170	0.171	0.326	0.839
beta2_pH[1,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[2,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[3,2]	-1.995	1.835	-6.798	-1.513	-0.030
beta2_pH[4,2]	-2.015	1.868	-6.752	-1.499	-0.026
beta2_pH[5,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[6,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[7,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[8,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[9,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[10,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[11,2]	-9.663	4.407	-20.699	-8.581	-4.116
beta2_pH[12,2]	-8.012	5.133	-20.194	-7.102	-0.906
beta2_pH[13,2]	-7.910	5.067	-20.265	-6.805	-1.730
beta2_pH[14,2]	-8.521	4.700	-20.356	-7.376	-2.512
beta2_pH[15,2]	-9.397	4.490	-20.668	-8.282	-3.721
beta2_pH[16,2]	-9.598	4.460	-20.695	-8.617	-3.910
beta2_pH[1,3]	0.242	0.261	0.101	0.181	0.700
beta2_pH[2,3]	0.000	0.000	0.000	0.000	0.000
beta2_pH[3,3]	0.000	0.000	0.000	0.000	0.000
beta2_pH[4,3]	0.000	0.000	0.000	0.000	0.000
beta2_pH[5,3]	9.189	6.428	-0.087	8.337	23.912
beta2_pH[6,3]	9.243	6.264	0.268	8.216	23.770
beta2_pH[7,3]	0.000	0.000	0.000	0.000	0.000
beta2_pH[8,3]	10.280	5.834	1.948	9.139	24.335
beta2_pH[9,3]	9.120	6.280	0.520	7.994	23.946
beta2_pH[10,3]	8.682	6.561	0.497	7.618	23.867
beta2_pH[11,3]	-2.211	1.973	-8.025	-1.634	-0.610
beta2_pH[12,3]	-2.396	1.863	-7.703	-1.862	-0.940
beta2_pH[13,3]	-2.826	2.266	-9.655	-2.083	-0.792
beta2_pH[14,3]	-2.759	2.134	-8.785	-2.071	-0.880
beta2_pH[15,3]	-2.911	2.240	-9.600	-2.168	-1.003
beta2_pH[16,3]	-2.979	2.294	-9.592	-2.216	-0.892
beta3_pH[1,1]	35.918	0.810	34.435	35.883	37.606
beta3_pH[2,1]	33.571	1.109	31.654	33.476	36.046
beta3_pH[3,1]	33.650	1.133	31.669	33.600	35.969
beta3_pH[4,1]	33.754	1.148	31.702	33.751	36.150
beta3_pH[5,1]	27.681	1.074	26.424	27.463	30.821
beta3_pH[6,1]	37.967	3.092	32.510	37.760	44.660
beta3_pH[7,1]	30.604	7.961	18.486	30.169	44.909
beta3_pH[8,1]	40.007	2.153	36.413	39.716	45.025
beta3_pH[9,1]	30.761	1.759	28.071	30.602	34.161
beta3_pH[10,1]	32.716	0.887	31.075	32.677	34.569
beta3_pH[11,1]	30.349	0.468	29.450	30.355	31.303
beta3_pH[12,1]	30.168	0.393	29.368	30.168	30.915
beta3_pH[13,1]	33.183	0.597	32.041	33.175	34.438
beta3_pH[14,1]	32.024	0.463	31.156	32.021	32.971
beta3_pH[15,1]	31.175	0.649	29.925	31.161	32.464
beta3_pH[16,1]	32.037	1.080	30.334	31.899	34.544
beta3_pH[1,2]	30.207	7.947	18.410	29.485	44.756
beta3_pH[2,2]	30.061	7.859	18.487	29.178	44.683
beta3_pH[3,2]	29.928	7.944	18.506	28.934	44.876
beta3_pH[4,2]	29.944	7.885	18.504	28.979	44.880
beta3_pH[5,2]	29.816	7.948	18.533	28.722	44.968
beta3_pH[6,2]	29.944	7.948	18.504	29.047	44.881
beta3_pH[7,2]	29.856	7.939	18.521	28.869	44.826
beta3_pH[8,2]	29.803	7.946	18.474	28.512	44.986
beta3_pH[9,2]	29.960	7.920	18.482	28.959	44.747
beta3_pH[10,2]	29.882	7.954	18.487	28.722	45.075
beta3_pH[11,2]	43.407	0.179	43.119	43.390	43.772
beta3_pH[12,2]	43.189	0.198	42.924	43.141	43.687
beta3_pH[13,2]	43.865	0.146	43.480	43.904	44.045
beta3_pH[14,2]	43.305	0.204	43.045	43.252	43.808
beta3_pH[15,2]	43.416	0.195	43.111	43.395	43.806
beta3_pH[16,2]	43.491	0.187	43.154	43.489	43.837
beta3_pH[1,3]	39.113	3.184	32.997	39.041	45.320
beta3_pH[2,3]	30.506	7.950	18.506	29.722	45.095
beta3_pH[3,3]	30.184	7.863	18.446	29.225	44.970
beta3_pH[4,3]	30.291	8.132	18.345	29.711	45.042
beta3_pH[5,3]	36.844	3.872	31.251	36.238	45.043
beta3_pH[6,3]	40.499	3.496	31.763	40.959	45.615
beta3_pH[7,3]	37.937	4.290	31.359	37.575	45.519
beta3_pH[8,3]	41.488	0.248	41.048	41.487	41.933
beta3_pH[9,3]	33.497	0.540	31.874	33.583	34.233
beta3_pH[10,3]	35.848	0.785	33.440	36.022	36.840
beta3_pH[11,3]	41.801	0.827	39.988	41.856	43.252
beta3_pH[12,3]	41.728	0.389	40.962	41.742	42.497
beta3_pH[13,3]	42.756	0.896	41.096	42.769	44.712
beta3_pH[14,3]	41.091	0.584	39.834	41.126	42.181
beta3_pH[15,3]	42.549	0.702	41.032	42.633	43.713
beta3_pH[16,3]	42.894	0.757	41.143	43.021	44.088
beta0_pelagic[1]	2.224	0.132	1.974	2.219	2.488
beta0_pelagic[2]	1.518	0.128	1.274	1.517	1.767
beta0_pelagic[3]	-0.609	0.769	-2.279	-0.504	0.500
beta0_pelagic[4]	-0.511	0.859	-2.638	-0.310	0.724
beta0_pelagic[5]	1.185	0.253	0.686	1.190	1.688
beta0_pelagic[6]	1.458	0.265	0.913	1.472	1.942
beta0_pelagic[7]	1.590	0.207	1.203	1.580	2.040
beta0_pelagic[8]	1.768	0.204	1.385	1.760	2.196
beta0_pelagic[9]	2.481	0.315	1.850	2.489	3.048
beta0_pelagic[10]	2.527	0.207	2.081	2.540	2.905
beta0_pelagic[11]	0.106	0.486	-1.399	0.217	0.702
beta0_pelagic[12]	1.681	0.144	1.399	1.686	1.955
beta0_pelagic[13]	0.296	0.213	-0.189	0.314	0.657
beta0_pelagic[14]	-0.097	0.269	-0.682	-0.073	0.357
beta0_pelagic[15]	-0.262	0.141	-0.537	-0.264	0.021
beta0_pelagic[16]	0.275	0.290	-0.478	0.343	0.667
beta1_pelagic[1]	0.000	0.000	0.000	0.000	0.000
beta1_pelagic[2]	0.000	0.000	0.000	0.000	0.000
beta1_pelagic[3]	2.428	1.324	0.600	2.164	5.319
beta1_pelagic[4]	2.008	1.080	0.537	1.776	4.661
beta1_pelagic[5]	-0.064	0.315	-0.707	-0.068	0.541
beta1_pelagic[6]	-0.092	0.447	-0.875	-0.124	0.741
beta1_pelagic[7]	-0.016	0.292	-0.585	-0.017	0.570
beta1_pelagic[8]	-0.017	0.285	-0.564	-0.020	0.561
beta1_pelagic[9]	0.202	0.486	-0.758	0.317	0.969
beta1_pelagic[10]	0.073	0.265	-0.441	0.065	0.604
beta1_pelagic[11]	3.642	1.179	2.166	3.421	6.521
beta1_pelagic[12]	2.773	0.324	2.197	2.761	3.413
beta1_pelagic[13]	2.961	0.751	1.793	2.879	4.760
beta1_pelagic[14]	4.330	1.072	2.753	4.138	6.832
beta1_pelagic[15]	2.910	0.267	2.386	2.914	3.422
beta1_pelagic[16]	3.633	0.939	2.665	3.303	6.398
beta2_pelagic[1]	0.000	0.000	0.000	0.000	0.000
beta2_pelagic[2]	0.000	0.000	0.000	0.000	0.000
beta2_pelagic[3]	0.558	2.069	0.032	0.108	6.194
beta2_pelagic[4]	1.103	3.000	0.026	0.285	11.093
beta2_pelagic[5]	-0.020	0.665	-1.406	-0.015	1.354
beta2_pelagic[6]	-0.099	0.683	-1.487	-0.138	1.359
beta2_pelagic[7]	-0.027	0.651	-1.414	-0.015	1.324
beta2_pelagic[8]	-0.003	0.655	-1.348	-0.010	1.436
beta2_pelagic[9]	0.200	0.693	-1.302	0.257	1.572
beta2_pelagic[10]	0.032	0.620	-1.309	0.037	1.372
beta2_pelagic[11]	1.938	3.994	0.106	0.305	14.160
beta2_pelagic[12]	6.141	5.075	0.978	4.648	19.655
beta2_pelagic[13]	0.926	2.018	0.172	0.441	6.105
beta2_pelagic[14]	0.337	0.287	0.161	0.290	0.746
beta2_pelagic[15]	6.490	5.124	1.227	5.049	21.020
beta2_pelagic[16]	4.773	5.557	0.195	3.030	20.001
beta3_pelagic[1]	29.640	7.892	18.439	28.405	44.784
beta3_pelagic[2]	29.837	7.982	18.465	28.876	44.812
beta3_pelagic[3]	30.254	7.269	18.853	29.305	44.698
beta3_pelagic[4]	25.208	5.492	18.477	24.095	40.687
beta3_pelagic[5]	30.177	8.313	18.506	28.770	45.313
beta3_pelagic[6]	31.846	6.789	19.035	31.597	44.357
beta3_pelagic[7]	29.814	7.843	18.418	28.926	44.822
beta3_pelagic[8]	29.849	8.038	18.482	28.661	44.733
beta3_pelagic[9]	30.806	6.062	19.303	31.039	42.597
beta3_pelagic[10]	29.413	8.100	18.418	28.153	44.794
beta3_pelagic[11]	42.586	1.826	37.683	43.018	45.571
beta3_pelagic[12]	43.472	0.290	42.987	43.460	43.996
beta3_pelagic[13]	42.828	1.317	40.424	42.809	45.487
beta3_pelagic[14]	42.386	1.655	39.099	42.333	45.564
beta3_pelagic[15]	43.182	0.256	42.552	43.182	43.676
beta3_pelagic[16]	43.079	0.909	40.618	43.191	45.069
mu_beta0_pelagic[1]	0.580	1.098	-1.861	0.613	2.791
mu_beta0_pelagic[2]	1.809	0.390	0.977	1.813	2.559
mu_beta0_pelagic[3]	0.331	0.459	-0.568	0.347	1.236
tau_beta0_pelagic[1]	0.402	0.433	0.050	0.258	1.541
tau_beta0_pelagic[2]	2.616	2.798	0.247	1.918	9.242
tau_beta0_pelagic[3]	1.529	1.171	0.178	1.258	4.476
beta0_yellow[1]	-0.539	0.197	-0.996	-0.522	-0.210
beta0_yellow[2]	0.498	0.170	0.137	0.507	0.796
beta0_yellow[3]	-0.327	0.200	-0.756	-0.314	0.024
beta0_yellow[4]	0.833	0.297	0.107	0.886	1.213
beta0_yellow[5]	-0.301	0.349	-0.993	-0.310	0.385
beta0_yellow[6]	1.115	0.166	0.790	1.115	1.450
beta0_yellow[7]	0.979	0.159	0.677	0.978	1.298
beta0_yellow[8]	1.012	0.152	0.710	1.013	1.309
beta0_yellow[9]	0.664	0.158	0.354	0.664	0.971
beta0_yellow[10]	0.578	0.142	0.303	0.576	0.863
beta0_yellow[11]	-1.940	0.427	-2.834	-1.938	-1.135
beta0_yellow[12]	-3.697	0.433	-4.651	-3.672	-2.919
beta0_yellow[13]	-3.692	0.456	-4.656	-3.667	-2.873
beta0_yellow[14]	-2.064	0.565	-3.038	-2.120	-0.541
beta0_yellow[15]	-2.856	0.403	-3.710	-2.834	-2.113
beta0_yellow[16]	-2.358	0.444	-3.211	-2.369	-1.437
beta1_yellow[1]	0.788	0.866	0.010	0.643	2.467
beta1_yellow[2]	1.077	0.373	0.584	1.019	2.040
beta1_yellow[3]	0.723	0.300	0.237	0.708	1.397
beta1_yellow[4]	1.404	0.811	0.654	1.184	3.960
beta1_yellow[5]	0.000	0.000	0.000	0.000	0.000
beta1_yellow[6]	0.000	0.000	0.000	0.000	0.000
beta1_yellow[7]	0.000	0.000	0.000	0.000	0.000
beta1_yellow[8]	0.000	0.000	0.000	0.000	0.000
beta1_yellow[9]	0.000	0.000	0.000	0.000	0.000
beta1_yellow[10]	0.000	0.000	0.000	0.000	0.000
beta1_yellow[11]	2.092	0.427	1.294	2.087	2.947
beta1_yellow[12]	2.490	0.448	1.716	2.466	3.482
beta1_yellow[13]	2.809	0.459	1.997	2.782	3.785
beta1_yellow[14]	2.162	0.529	0.886	2.187	3.139
beta1_yellow[15]	2.107	0.399	1.384	2.089	2.939
beta1_yellow[16]	2.113	0.445	1.185	2.129	2.941
beta2_yellow[1]	-3.630	3.126	-11.239	-2.876	-0.041
beta2_yellow[2]	-3.625	3.118	-11.505	-2.766	-0.178
beta2_yellow[3]	-3.435	3.036	-10.921	-2.640	-0.123
beta2_yellow[4]	-3.015	3.098	-10.849	-2.013	-0.096
beta2_yellow[5]	0.000	0.000	0.000	0.000	0.000
beta2_yellow[6]	0.000	0.000	0.000	0.000	0.000
beta2_yellow[7]	0.000	0.000	0.000	0.000	0.000
beta2_yellow[8]	0.000	0.000	0.000	0.000	0.000
beta2_yellow[9]	0.000	0.000	0.000	0.000	0.000
beta2_yellow[10]	0.000	0.000	0.000	0.000	0.000
beta2_yellow[11]	-4.402	2.494	-10.361	-3.912	-1.038
beta2_yellow[12]	-4.844	2.700	-11.796	-4.219	-1.440
beta2_yellow[13]	-4.391	2.258	-10.373	-3.899	-1.429
beta2_yellow[14]	-4.640	2.810	-11.643	-4.119	-0.250
beta2_yellow[15]	-4.230	2.622	-10.967	-3.569	-0.966
beta2_yellow[16]	-4.877	2.701	-11.963	-4.260	-1.369
beta3_yellow[1]	25.774	7.095	18.252	22.644	43.956
beta3_yellow[2]	29.066	1.818	25.182	28.889	32.807
beta3_yellow[3]	32.888	3.123	25.093	32.834	39.830
beta3_yellow[4]	28.872	3.452	21.320	27.974	35.986
beta3_yellow[5]	29.841	7.815	18.473	28.660	44.804
beta3_yellow[6]	29.956	7.852	18.497	28.985	44.566
beta3_yellow[7]	30.014	7.799	18.451	29.299	44.897
beta3_yellow[8]	29.859	7.881	18.425	28.989	44.740
beta3_yellow[9]	30.038	7.982	18.419	29.264	44.781
beta3_yellow[10]	30.303	7.999	18.483	29.608	44.965
beta3_yellow[11]	45.284	0.543	43.990	45.376	45.974
beta3_yellow[12]	43.306	0.376	42.543	43.290	44.026
beta3_yellow[13]	44.835	0.391	43.981	44.907	45.501
beta3_yellow[14]	43.861	2.079	35.188	44.192	45.839
beta3_yellow[15]	45.158	0.521	44.139	45.147	45.958
beta3_yellow[16]	44.565	0.646	43.427	44.547	45.863
mu_beta0_yellow[1]	0.095	0.556	-1.075	0.088	1.200
mu_beta0_yellow[2]	0.637	0.336	-0.115	0.659	1.238
mu_beta0_yellow[3]	-2.415	0.630	-3.393	-2.505	-0.945
tau_beta0_yellow[1]	1.692	1.869	0.093	1.154	6.666
tau_beta0_yellow[2]	3.464	3.847	0.310	2.420	13.260
tau_beta0_yellow[3]	1.296	1.596	0.104	0.852	5.005
beta0_black[1]	-0.077	0.161	-0.394	-0.079	0.235
beta0_black[2]	1.912	0.128	1.661	1.913	2.170
beta0_black[3]	1.318	0.137	1.047	1.318	1.586
beta0_black[4]	2.428	0.131	2.172	2.425	2.689
beta0_black[5]	4.611	2.082	1.766	4.179	9.781
beta0_black[6]	4.650	1.933	2.270	4.179	9.823
beta0_black[7]	3.759	1.875	1.537	3.271	9.181
beta0_black[8]	0.959	0.212	0.555	0.958	1.378
beta0_black[9]	2.609	0.230	2.157	2.604	3.063
beta0_black[10]	1.457	0.133	1.201	1.457	1.716
beta0_black[11]	3.481	0.155	3.195	3.478	3.792
beta0_black[12]	4.865	0.174	4.529	4.865	5.208
beta0_black[13]	-0.139	0.270	-0.688	-0.126	0.332
beta0_black[14]	2.849	0.161	2.529	2.847	3.166
beta0_black[15]	1.297	0.154	0.996	1.297	1.598
beta0_black[16]	4.274	0.164	3.956	4.271	4.606
beta2_black[1]	7.790	9.952	0.539	3.554	38.408
beta2_black[2]	0.000	0.000	0.000	0.000	0.000
beta2_black[3]	0.000	0.000	0.000	0.000	0.000
beta2_black[4]	0.000	0.000	0.000	0.000	0.000
beta2_black[5]	0.000	0.000	0.000	0.000	0.000
beta2_black[6]	0.000	0.000	0.000	0.000	0.000
beta2_black[7]	0.000	0.000	0.000	0.000	0.000
beta2_black[8]	0.000	0.000	0.000	0.000	0.000
beta2_black[9]	0.000	0.000	0.000	0.000	0.000
beta2_black[10]	0.000	0.000	0.000	0.000	0.000
beta2_black[11]	0.000	0.000	0.000	0.000	0.000
beta2_black[12]	0.000	0.000	0.000	0.000	0.000
beta2_black[13]	-1.751	1.512	-6.081	-1.283	-0.272
beta2_black[14]	0.000	0.000	0.000	0.000	0.000
beta2_black[15]	0.000	0.000	0.000	0.000	0.000
beta2_black[16]	0.000	0.000	0.000	0.000	0.000
beta3_black[1]	41.802	1.097	39.778	41.959	43.367
beta3_black[2]	25.000	0.000	25.000	25.000	25.000
beta3_black[3]	25.000	0.000	25.000	25.000	25.000
beta3_black[4]	25.000	0.000	25.000	25.000	25.000
beta3_black[5]	25.000	0.000	25.000	25.000	25.000
beta3_black[6]	25.000	0.000	25.000	25.000	25.000
beta3_black[7]	25.000	0.000	25.000	25.000	25.000
beta3_black[8]	25.000	0.000	25.000	25.000	25.000
beta3_black[9]	25.000	0.000	25.000	25.000	25.000
beta3_black[10]	25.000	0.000	25.000	25.000	25.000
beta3_black[11]	25.000	0.000	25.000	25.000	25.000
beta3_black[12]	25.000	0.000	25.000	25.000	25.000
beta3_black[13]	39.072	1.410	36.724	39.267	40.654
beta3_black[14]	25.000	0.000	25.000	25.000	25.000
beta3_black[15]	25.000	0.000	25.000	25.000	25.000
beta3_black[16]	25.000	0.000	25.000	25.000	25.000
beta4_black[1]	-0.259	0.194	-0.640	-0.261	0.120
beta4_black[2]	0.247	0.185	-0.122	0.248	0.614
beta4_black[3]	-0.932	0.196	-1.325	-0.934	-0.542
beta4_black[4]	0.425	0.220	-0.005	0.423	0.855
beta4_black[5]	0.544	1.438	-1.223	0.309	3.702
beta4_black[6]	0.543	1.410	-1.286	0.317	3.698
beta4_black[7]	0.421	1.200	-1.410	0.257	3.199
beta4_black[8]	-0.247	0.314	-0.883	-0.237	0.353
beta4_black[9]	0.821	0.768	-0.272	0.680	2.722
beta4_black[10]	0.051	0.185	-0.320	0.053	0.420
beta4_black[11]	-0.689	0.214	-1.107	-0.684	-0.280
beta4_black[12]	0.169	0.325	-0.441	0.162	0.821
beta4_black[13]	-1.190	0.222	-1.636	-1.183	-0.762
beta4_black[14]	-0.176	0.246	-0.652	-0.180	0.309
beta4_black[15]	-0.895	0.216	-1.322	-0.896	-0.470
beta4_black[16]	-0.595	0.234	-1.046	-0.595	-0.133
mu_beta0_black[1]	1.289	0.884	-0.630	1.326	2.986
mu_beta0_black[2]	2.700	1.061	0.774	2.589	5.086
mu_beta0_black[3]	2.476	1.004	0.336	2.536	4.365
tau_beta0_black[1]	0.635	0.588	0.056	0.454	2.157
tau_beta0_black[2]	0.463	0.637	0.046	0.252	2.203
tau_beta0_black[3]	0.230	0.152	0.050	0.191	0.626
beta0_dsr[11]	-2.888	0.295	-3.495	-2.888	-2.327
beta0_dsr[12]	4.554	0.278	3.995	4.557	5.090
beta0_dsr[13]	-1.368	0.355	-2.015	-1.355	-0.774
beta0_dsr[14]	-3.659	0.517	-4.626	-3.658	-2.620
beta0_dsr[15]	-1.943	0.279	-2.490	-1.949	-1.401
beta0_dsr[16]	-3.000	0.369	-3.727	-2.986	-2.303
beta1_dsr[11]	4.825	0.307	4.235	4.821	5.421
beta1_dsr[12]	10.707	55.183	2.282	5.049	22.960
beta1_dsr[13]	2.894	0.428	2.253	2.864	3.604
beta1_dsr[14]	6.328	0.543	5.230	6.339	7.346
beta1_dsr[15]	3.339	0.283	2.788	3.338	3.893
beta1_dsr[16]	5.814	0.383	5.080	5.817	6.564
beta2_dsr[11]	-8.345	2.402	-14.088	-8.003	-4.710
beta2_dsr[12]	-7.141	2.620	-12.849	-6.968	-2.503
beta2_dsr[13]	-6.403	2.844	-12.529	-6.348	-0.701
beta2_dsr[14]	-6.147	2.714	-11.938	-5.947	-1.875
beta2_dsr[15]	-7.783	2.440	-13.347	-7.463	-3.925
beta2_dsr[16]	-7.982	2.365	-13.359	-7.630	-4.264
beta3_dsr[11]	43.487	0.152	43.210	43.484	43.767
beta3_dsr[12]	33.973	0.728	32.154	34.115	34.810
beta3_dsr[13]	43.248	0.359	42.740	43.195	43.896
beta3_dsr[14]	43.347	0.240	43.071	43.278	43.964
beta3_dsr[15]	43.503	0.185	43.172	43.505	43.843
beta3_dsr[16]	43.441	0.163	43.164	43.426	43.760
beta4_dsr[11]	0.582	0.220	0.162	0.579	1.024
beta4_dsr[12]	0.231	0.441	-0.659	0.226	1.088
beta4_dsr[13]	-0.169	0.222	-0.598	-0.166	0.256
beta4_dsr[14]	0.146	0.252	-0.359	0.145	0.644
beta4_dsr[15]	0.726	0.217	0.325	0.716	1.166
beta4_dsr[16]	0.152	0.226	-0.280	0.153	0.593
beta0_slope[11]	-1.847	0.150	-2.139	-1.846	-1.560
beta0_slope[12]	-4.464	0.254	-4.970	-4.460	-3.966
beta0_slope[13]	-1.342	0.182	-1.746	-1.328	-1.037
beta0_slope[14]	-2.678	0.165	-3.008	-2.674	-2.368
beta0_slope[15]	-1.340	0.146	-1.629	-1.341	-1.056
beta0_slope[16]	-2.735	0.154	-3.027	-2.740	-2.427
beta1_slope[11]	4.486	0.225	4.051	4.490	4.916
beta1_slope[12]	3.980	0.448	3.108	3.978	4.843
beta1_slope[13]	2.714	0.440	2.206	2.647	3.972
beta1_slope[14]	6.323	0.413	5.519	6.319	7.124
beta1_slope[15]	3.009	0.210	2.593	3.010	3.414
beta1_slope[16]	5.286	0.284	4.738	5.283	5.849
beta2_slope[11]	8.551	2.267	5.055	8.274	13.768
beta2_slope[12]	6.600	2.858	1.218	6.647	12.303
beta2_slope[13]	5.264	3.015	0.395	5.120	11.339
beta2_slope[14]	6.325	2.554	2.199	6.115	11.829
beta2_slope[15]	8.122	2.328	4.560	7.843	13.561
beta2_slope[16]	7.648	2.275	4.312	7.287	13.271
beta3_slope[11]	43.459	0.134	43.225	43.455	43.734
beta3_slope[12]	43.355	0.289	42.900	43.305	43.925
beta3_slope[13]	43.457	0.393	42.879	43.407	44.059
beta3_slope[14]	43.271	0.139	43.094	43.236	43.623
beta3_slope[15]	43.490	0.164	43.195	43.488	43.802
beta3_slope[16]	43.370	0.140	43.154	43.350	43.687
beta4_slope[11]	-0.730	0.164	-1.069	-0.727	-0.413
beta4_slope[12]	-1.151	0.466	-2.153	-1.107	-0.352
beta4_slope[13]	0.091	0.163	-0.231	0.095	0.415
beta4_slope[14]	-0.095	0.197	-0.467	-0.096	0.283
beta4_slope[15]	-0.759	0.159	-1.078	-0.756	-0.450
beta4_slope[16]	-0.168	0.175	-0.531	-0.167	0.175
sigma_H[1]	0.199	0.052	0.101	0.198	0.311
sigma_H[2]	0.172	0.030	0.120	0.170	0.239
sigma_H[3]	0.196	0.042	0.122	0.192	0.285
sigma_H[4]	0.419	0.075	0.296	0.411	0.584
sigma_H[5]	0.997	0.209	0.620	0.987	1.437
sigma_H[6]	0.392	0.202	0.031	0.384	0.817
sigma_H[7]	0.307	0.062	0.212	0.300	0.451
sigma_H[8]	0.416	0.094	0.279	0.405	0.610
sigma_H[9]	0.529	0.127	0.330	0.513	0.821
sigma_H[10]	0.215	0.041	0.144	0.212	0.304
sigma_H[11]	0.277	0.045	0.199	0.274	0.376
sigma_H[12]	0.439	0.166	0.208	0.416	0.782
sigma_H[13]	0.216	0.039	0.149	0.212	0.301
sigma_H[14]	0.508	0.091	0.346	0.502	0.698
sigma_H[15]	0.246	0.040	0.180	0.242	0.335
sigma_H[16]	0.223	0.042	0.155	0.219	0.317
lambda_H[1]	3.179	4.326	0.146	1.829	13.481
lambda_H[2]	8.332	8.276	0.836	6.063	28.086
lambda_H[3]	6.128	9.167	0.263	3.019	30.426
lambda_H[4]	0.006	0.004	0.001	0.005	0.018
lambda_H[5]	4.365	11.477	0.038	1.128	30.304
lambda_H[6]	7.916	16.405	0.008	0.817	59.636
lambda_H[7]	0.013	0.009	0.002	0.011	0.036
lambda_H[8]	8.299	10.639	0.135	4.620	36.539
lambda_H[9]	0.015	0.010	0.003	0.013	0.040
lambda_H[10]	0.321	0.747	0.036	0.206	1.192
lambda_H[11]	0.262	0.395	0.011	0.128	1.358
lambda_H[12]	4.590	5.951	0.181	2.671	20.462
lambda_H[13]	3.509	3.165	0.240	2.658	11.681
lambda_H[14]	3.289	3.805	0.224	2.077	12.447
lambda_H[15]	0.025	0.044	0.003	0.016	0.095
lambda_H[16]	0.846	1.354	0.044	0.441	4.024
mu_lambda_H[1]	4.319	1.946	1.171	4.159	8.480
mu_lambda_H[2]	3.826	1.957	0.604	3.667	8.082
mu_lambda_H[3]	3.439	1.819	0.780	3.122	7.729
sigma_lambda_H[1]	8.577	4.380	2.035	7.973	18.290
sigma_lambda_H[2]	8.327	4.723	0.974	7.789	18.396
sigma_lambda_H[3]	6.150	3.932	1.060	5.297	16.662
beta_H[1,1]	6.914	1.083	4.333	7.073	8.529
beta_H[2,1]	9.888	0.483	8.809	9.906	10.817
beta_H[3,1]	7.979	0.776	6.046	8.094	9.204
beta_H[4,1]	9.548	7.934	-6.325	9.736	25.235
beta_H[5,1]	0.149	2.305	-4.626	0.358	4.078
beta_H[6,1]	3.135	3.990	-6.905	4.509	7.627
beta_H[7,1]	0.586	5.697	-11.567	0.866	10.736
beta_H[8,1]	1.292	3.450	-2.484	1.229	3.563
beta_H[9,1]	13.139	5.730	1.961	13.029	24.654
beta_H[10,1]	7.058	1.653	3.636	7.124	10.256
beta_H[11,1]	5.071	3.500	-2.871	5.913	9.914
beta_H[12,1]	2.624	1.069	0.686	2.566	4.998
beta_H[13,1]	9.054	0.850	7.255	9.118	10.391
beta_H[14,1]	2.168	1.001	0.179	2.163	4.087
beta_H[15,1]	-6.191	3.733	-12.856	-6.417	1.815
beta_H[16,1]	3.467	2.708	-0.862	3.169	9.833
beta_H[1,2]	7.909	0.239	7.423	7.913	8.354
beta_H[2,2]	10.027	0.134	9.770	10.025	10.284
beta_H[3,2]	8.950	0.198	8.567	8.949	9.353
beta_H[4,2]	3.545	1.472	0.747	3.503	6.471
beta_H[5,2]	1.953	0.972	-0.036	1.976	3.778
beta_H[6,2]	5.719	1.080	3.201	5.908	7.349
beta_H[7,2]	2.660	1.095	0.690	2.590	4.994
beta_H[8,2]	3.008	1.046	1.392	3.124	4.215
beta_H[9,2]	3.463	1.103	1.349	3.433	5.676
beta_H[10,2]	8.207	0.342	7.495	8.216	8.861
beta_H[11,2]	9.773	0.632	8.849	9.645	11.197
beta_H[12,2]	3.936	0.380	3.178	3.929	4.707
beta_H[13,2]	9.123	0.247	8.654	9.115	9.635
beta_H[14,2]	4.021	0.357	3.328	4.008	4.742
beta_H[15,2]	11.376	0.671	9.938	11.405	12.623
beta_H[16,2]	4.520	0.800	3.015	4.508	6.146
beta_H[1,3]	8.459	0.234	8.039	8.445	8.957
beta_H[2,3]	10.063	0.116	9.834	10.063	10.295
beta_H[3,3]	9.615	0.163	9.296	9.613	9.954
beta_H[4,3]	-2.508	0.878	-4.227	-2.493	-0.835
beta_H[5,3]	3.824	0.614	2.541	3.847	4.990
beta_H[6,3]	8.022	1.190	6.379	7.644	10.613
beta_H[7,3]	-2.802	0.654	-4.089	-2.796	-1.579
beta_H[8,3]	5.250	0.480	4.642	5.194	6.219
beta_H[9,3]	-2.836	0.730	-4.282	-2.830	-1.448
beta_H[10,3]	8.685	0.271	8.161	8.681	9.219
beta_H[11,3]	8.536	0.287	7.927	8.556	9.047
beta_H[12,3]	5.250	0.322	4.464	5.293	5.775
beta_H[13,3]	8.846	0.175	8.483	8.851	9.178
beta_H[14,3]	5.713	0.277	5.103	5.733	6.192
beta_H[15,3]	10.370	0.314	9.767	10.365	11.007
beta_H[16,3]	6.241	0.597	4.909	6.302	7.255
beta_H[1,4]	8.263	0.181	7.874	8.272	8.598
beta_H[2,4]	10.130	0.119	9.879	10.136	10.348
beta_H[3,4]	10.114	0.167	9.747	10.126	10.411
beta_H[4,4]	11.795	0.444	10.909	11.789	12.686
beta_H[5,4]	5.461	0.736	4.274	5.368	7.177
beta_H[6,4]	7.031	0.943	4.944	7.322	8.313
beta_H[7,4]	8.289	0.352	7.597	8.290	8.994
beta_H[8,4]	6.717	0.248	6.246	6.726	7.143
beta_H[9,4]	7.204	0.472	6.273	7.197	8.126
beta_H[10,4]	7.760	0.233	7.325	7.747	8.240
beta_H[11,4]	9.389	0.201	8.998	9.391	9.779
beta_H[12,4]	7.144	0.214	6.758	7.136	7.596
beta_H[13,4]	9.041	0.145	8.746	9.048	9.311
beta_H[14,4]	7.727	0.222	7.289	7.721	8.178
beta_H[15,4]	9.468	0.237	9.020	9.467	9.940
beta_H[16,4]	9.338	0.237	8.902	9.325	9.830
beta_H[1,5]	8.994	0.145	8.696	8.998	9.273
beta_H[2,5]	10.786	0.093	10.610	10.784	10.968
beta_H[3,5]	10.922	0.174	10.618	10.909	11.293
beta_H[4,5]	8.397	0.458	7.544	8.378	9.367
beta_H[5,5]	5.424	0.563	4.107	5.461	6.412
beta_H[6,5]	8.798	0.628	7.898	8.646	10.301
beta_H[7,5]	6.739	0.342	6.076	6.736	7.424
beta_H[8,5]	8.214	0.209	7.849	8.206	8.640
beta_H[9,5]	8.199	0.476	7.270	8.199	9.155
beta_H[10,5]	10.086	0.227	9.609	10.086	10.517
beta_H[11,5]	11.506	0.225	11.063	11.508	11.944
beta_H[12,5]	8.482	0.200	8.085	8.480	8.895
beta_H[13,5]	10.007	0.130	9.758	10.006	10.272
beta_H[14,5]	9.199	0.238	8.772	9.186	9.693
beta_H[15,5]	11.174	0.245	10.680	11.175	11.642
beta_H[16,5]	9.912	0.180	9.548	9.914	10.255
beta_H[1,6]	10.176	0.187	9.841	10.163	10.575
beta_H[2,6]	11.513	0.108	11.303	11.511	11.730
beta_H[3,6]	10.814	0.163	10.454	10.826	11.104
beta_H[4,6]	12.876	0.806	11.196	12.892	14.378
beta_H[5,6]	5.879	0.592	4.726	5.862	7.040
beta_H[6,6]	8.764	0.688	6.960	8.888	9.738
beta_H[7,6]	9.889	0.572	8.819	9.884	11.044
beta_H[8,6]	9.519	0.273	9.041	9.529	9.947
beta_H[9,6]	8.466	0.789	6.993	8.450	10.090
beta_H[10,6]	9.514	0.313	8.845	9.545	10.064
beta_H[11,6]	10.810	0.352	10.040	10.839	11.437
beta_H[12,6]	9.376	0.260	8.867	9.366	9.909
beta_H[13,6]	11.048	0.169	10.746	11.041	11.411
beta_H[14,6]	9.820	0.291	9.237	9.829	10.364
beta_H[15,6]	10.829	0.431	9.988	10.830	11.690
beta_H[16,6]	10.522	0.240	10.012	10.530	10.988
beta_H[1,7]	10.864	0.862	8.679	10.965	12.249
beta_H[2,7]	12.215	0.430	11.306	12.225	13.039
beta_H[3,7]	10.535	0.660	9.071	10.579	11.657
beta_H[4,7]	2.520	4.189	-5.213	2.381	11.037
beta_H[5,7]	6.389	1.752	3.013	6.359	10.299
beta_H[6,7]	9.565	2.507	4.498	9.518	16.106
beta_H[7,7]	10.411	2.871	4.764	10.444	16.006
beta_H[8,7]	10.942	0.983	9.380	10.902	12.560
beta_H[9,7]	4.515	4.005	-3.644	4.634	12.152
beta_H[10,7]	9.813	1.449	7.155	9.703	12.949
beta_H[11,7]	11.032	1.715	7.866	10.888	14.885
beta_H[12,7]	9.983	0.987	7.756	10.064	11.564
beta_H[13,7]	11.652	0.767	9.921	11.743	12.872
beta_H[14,7]	10.381	0.946	8.342	10.444	12.033
beta_H[15,7]	12.040	2.252	7.556	12.047	16.338
beta_H[16,7]	12.302	1.255	10.174	12.142	15.106
beta0_H[1]	8.626	13.139	-18.178	8.631	35.204
beta0_H[2]	10.612	6.386	-3.235	10.709	24.160
beta0_H[3]	9.719	10.295	-9.875	9.799	28.883
beta0_H[4]	10.321	180.823	-346.937	7.579	403.966
beta0_H[5]	4.317	24.554	-40.809	4.254	51.540
beta0_H[6]	5.769	55.249	-117.433	7.387	119.390
beta0_H[7]	5.252	134.320	-271.868	2.121	264.322
beta0_H[8]	6.333	25.454	-16.532	6.397	25.615
beta0_H[9]	5.156	124.068	-242.427	5.481	243.516
beta0_H[10]	8.525	31.676	-57.019	8.262	73.555
beta0_H[11]	10.674	50.092	-95.814	11.119	120.251
beta0_H[12]	6.902	11.532	-16.680	6.842	29.365
beta0_H[13]	10.035	10.925	-10.475	9.893	30.304
beta0_H[14]	7.018	11.553	-15.812	7.024	31.098
beta0_H[15]	8.636	106.948	-207.834	8.119	218.083
beta0_H[16]	8.501	25.204	-44.559	8.802	59.810

## [1] 10